Free AIME Combinatorics Questions and Answers — Questions and Answers
Question 1: How many ways can 5 people be seated in a row?
- 24
- 60
- 120 (Correct answer)
- 720
Correct answer: 120
The number of ways to arrange 𝑛 people in a row is 𝑛! n!. For 5 people, this is 5!=5×4×3×2×1=120.
Question 2: In how many ways can a committee of 3 people be chosen from a group of 10?
- 30
- 45
- 120 (Correct answer)
- 720
Correct answer: 120
The number of ways to choose 𝑘 people from 𝑛 is given by 120
Question 3: How many ways can the letters of the word "AIME" be arranged?
- 12
- 24 (Correct answer)
- 120
- 720
Correct answer: 24
The number of ways to arrange the letters of a word with all distinct letters is 𝑛! For "AIME", which has 4 letters, the number of arrangements is 4!=4×3×2×1=24.
Question 4: How many ways can you distribute 5 identical candies to 3 children so that each child gets at least one candy?
- 6
- 21
- 15
- 10 (Correct answer)
Correct answer: 10
This is a problem of distributing indistinguishable objects (candies) into distinguishable bins (children) with the restriction that each bin gets at least one object.<br> This is solved using the stars and bars method. We first give each child one candy, then distribute the remaining 2 candies among the 3 children. The number of ways is given by 6.
Question 5: In how many ways can you choose 2 different cards from a standard deck of 52 cards?
- 1326 (Correct answer)
- 1378
- 1324
- 1348
Correct answer: 1326
The number of ways to choose 2 cards from 52 is given by 1326
Free AIME Combinatorics Questions and Answers
How many ways can 5 people be seated in a row?